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Tag 02X3

53.13. Separation conditions on algebraic spaces

A separation condition on an algebraic space $F$ is a condition on the diagonal morphism $F \to F \times F$. Let us first list the properties the diagonal has automatically. Since the diagonal is representable by definition the following lemma makes sense (through Definition 53.5.1).

Lemma 53.13.1. Let $S$ be a scheme contained in $\textit{Sch}_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism. Then

  1. $\Delta$ is locally of finite type,
  2. $\Delta$ is a monomorphism,
  3. $\Delta$ is separated, and
  4. $\Delta$ is locally quasi-finite.

Proof. Let $F = U/R$ be a presentation of $F$. As in the proof of Lemma 53.10.4 the diagram $$ \xymatrix{ R \ar[r] \ar[d]_j & F \ar[d]^\Delta \\ U \times_S U \ar[r] & F \times F } $$ is cartesian. Hence according to Lemma 53.11.4 it suffices to show that $j$ has the properties listed in the lemma. (Note that each of the properties (1) – (4) occur in the lists of Remarks 53.4.1 and 53.4.3.) Since $j$ is an equivalence relation it is a monomorphism. Hence it is separated by Schemes, Lemma 25.23.3. As $R$ is an étale equivalence relation we see that $s, t : R \to U$ are étale. Hence $s, t$ are locally of finite type. Then it follows from Morphisms, Lemma 28.14.8 that $j$ is locally of finite type. Finally, as it is a monomorphism its fibres are finite. Thus we conclude that it is locally quasi-finite by Morphisms, Lemma 28.19.7. $\square$

Here are some common types of separation conditions, relative to the base scheme $S$. There is also an absolute notion of these conditions which we will discuss in Properties of Spaces, Section 54.3. Moreover, we will discuss separation conditions for a morphism of algebraic spaces in Morphisms of Spaces, Section 55.4.

Definition 53.13.2. Let $S$ be a scheme contained in $\textit{Sch}_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism.

  1. We say $F$ is separated over $S$ if $\Delta$ is a closed immersion.
  2. We say $F$ is locally separated over $S$1 if $\Delta$ is an immersion.
  3. We say $F$ is quasi-separated over $S$ if $\Delta$ is quasi-compact.
  4. We say $F$ is Zariski locally quasi-separated over $S$2 if there exists a Zariski covering $F = \bigcup_{i \in I} F_i$ such that each $F_i$ is quasi-separated.

Note that if the diagonal is quasi-compact (when $F$ is separated or quasi-separated) then the diagonal is actually quasi-finite and separated, hence quasi-affine (by More on Morphisms, Lemma 36.38.2).

  1. In the literature this often refers to quasi-separated and locally separated algebraic spaces.
  2. This definition was suggested by B. Conrad.

The code snippet corresponding to this tag is a part of the file spaces.tex and is located in lines 2038–2130 (see updates for more information).

\section{Separation conditions on algebraic spaces}
\label{section-separation}

\noindent
A separation condition on an algebraic space $F$ is a condition
on the diagonal morphism $F \to F \times F$. Let us first
list the properties the diagonal has automatically.
Since the diagonal is representable by definition the following lemma
makes sense (through
Definition \ref{definition-relative-representable-property}).

\begin{lemma}
\label{lemma-properties-diagonal}
Let $S$ be a scheme contained in $\Sch_{fppf}$.
Let $F$ be an algebraic space over $S$.
Let $\Delta : F \to F \times F$ be the diagonal morphism.
Then
\begin{enumerate}
\item $\Delta$ is locally of finite type,
\item $\Delta$ is a monomorphism,
\item $\Delta$ is separated, and
\item $\Delta$ is locally quasi-finite.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $F = U/R$ be a presentation of $F$.
As in the proof of Lemma \ref{lemma-presentation-quasi-compact} the diagram
$$
\xymatrix{
R \ar[r] \ar[d]_j & F \ar[d]^\Delta \\
U \times_S U \ar[r] & F \times F
}
$$
is cartesian. Hence according to
Lemma \ref{lemma-representable-morphisms-spaces-property}
it suffices to show that $j$ has the properties listed in the lemma.
(Note that each of the properties (1) -- (4) occur in the lists
of Remarks \ref{remark-list-properties-stable-base-change}
and \ref{remark-list-properties-fpqc-local-base}.)
Since $j$ is an equivalence relation it is a monomorphism.
Hence it is separated by
Schemes, Lemma \ref{schemes-lemma-monomorphism-separated}.
As $R$ is an \'etale equivalence relation we see that
$s, t : R \to U$ are \'etale. Hence $s, t$ are locally of finite
type. Then it follows from
Morphisms, Lemma \ref{morphisms-lemma-permanence-finite-type} that
$j$ is locally of finite type. Finally, as it is a monomorphism
its fibres are finite. Thus we conclude that it is locally quasi-finite by
Morphisms, Lemma \ref{morphisms-lemma-finite-fibre}.
\end{proof}

\noindent
Here are some common types of separation conditions, relative to the base
scheme $S$. There is also an absolute notion of these conditions which we
will discuss in
Properties of Spaces, Section \ref{spaces-properties-section-separation}.
Moreover, we will discuss separation conditions for a morphism of
algebraic spaces in
Morphisms of Spaces, Section \ref{spaces-morphisms-section-separation-axioms}.

\begin{definition}
\label{definition-separated}
Let $S$ be a scheme contained in $\Sch_{fppf}$.
Let $F$ be an algebraic space over $S$.
Let $\Delta : F \to F \times F$ be the diagonal morphism.
\begin{enumerate}
\item We say $F$ is {\it separated over $S$} if $\Delta$ is a closed immersion.
\item We say $F$ is {\it locally separated over $S$}\footnote{In the
literature this often refers to quasi-separated and
locally separated algebraic spaces.} if $\Delta$ is an
immersion.
\item We say $F$ is {\it quasi-separated over $S$} if $\Delta$ is quasi-compact.
\item We say $F$ is {\it Zariski locally quasi-separated over $S$}\footnote{This
definition was suggested by B.\ Conrad.} if there
exists a Zariski covering $F = \bigcup_{i \in I} F_i$ such that
each $F_i$ is quasi-separated.
\end{enumerate}
\end{definition}

\noindent
Note that if the diagonal is quasi-compact (when $F$ is separated or
quasi-separated) then the diagonal is actually
quasi-finite and separated, hence quasi-affine (by More on Morphisms,
Lemma \ref{more-morphisms-lemma-quasi-finite-separated-quasi-affine}).

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